generalized solutions of variational problems and...

GENERALIZED SOLUTIONS OF VARIATIONAL

PROBLEMS AND APPLICATIONS

VIERI BENCI, LORENZO LUPERI BAGLINI, AND MARCO SQUASSINA

Abstract. Ultrafunctions are a particular class of generalized functions defined on a hyperreal fieldR∗ ⊃ R that allow to solve variational problems with no classical solutions. We recall the construction of

ultrafunctions and we study the relationships between these generalized solutions and classical minimizing

sequences. Finally, we study some examples to highlight the potential of this approach.

Contents

1. Introduction 11.1. Notations 22. Λ-theory 32.1. Non Archimedean Fields 32.2. The Λ-limit 32.3. Natural extension of sets and functions 52.4. Hyperfinite sets and hyperfinite sums 73. Ultrafunctions 73.1. Definition of ultrafunctions 73.2. The splitting of an ultrafunction 103.3. The Gauss divergence theorem 123.4. Ultrafunctions and distributions 134. Properties of ultrafunction solutions 145. Applications 175.1. Sign-perturbation of potentials 175.2. The singular variational problem 19References 22

1. Introduction

It is nowadays very well known that, in many circumstances, the needs of a theory require the introductionof generalized functions. Among people working in partial differential equations, the theory of distributionsof L. Schwartz is the most commonly used, but other notions of generalized functions have been introduced,e.g. by J.F. Colombeau [15] and M. Sato [21, 22]. Many notions of generalized functions are based onnon-Archimedean mathematics, namely mathematics handling infinite and/or infinitesimal quantities.Such an approach presents several positive features, the main probably being the possibility of treatingdistributions as non-Archimedean set-theoretical functions (under the limitations imposed by Schwartz’result). This allows to easily introduce nonlinear concepts, such as products, into distribution theory.Moreover, a theory which includes infinitesimals and infinite quantities makes it possible to easily constructnew models, allowing in this way to study several problems which are difficult even to formalize in classicalmathematics. This has led to applications in various field, including several topics in analysis, geometryand mathematical physic (see e.g. [16, 18] for an overview in the case of Colombeau functions and theirrecent extension, called generalized smooth functions).

2010 Mathematics Subject Classification. 03H05, 26E35, 28E05, 46S20.

Key words and phrases. Ultrafunctions, Non Archimedean Mathematics, Non Standard Analysis, Delta function.M. Squassina is a member of the Gruppo Nazionale per l’Analisi Matematica, la Probabilita e le loro Applicazioni

(GNAMPA) and of Istituto Nazionale di Alta Matematica (INdAM). L. Luperi Baglini has been supported by grants

M1876-N35, P26859-N25 and P 30821-N35 of the Austrian Science Fund FWF.

1

2 VIERI BENCI, LORENZO LUPERI BAGLINI, AND MARCO SQUASSINA

In this paper we deal with ultrafunctions, which are a kind of generalized functions that have beenintroduced recently in [1] and developed in [2, 4–11].

Ultrafunctions are a particular case of non-Archimedean generalized functions that are based on thehyperreal field R∗, namely the numerical field on which nonstandard analysis1 is based. No prior knowledgeof nonstandard analysis is requested to read this paper: we will introduce all the nonstandard notions thatwe need via a new notion of limit, called Λ-limit (see [10] for a complete introduction to this notion and itsrelationships with the usual nonstandard analysis). The main peculiarity of this notion of limit is that itallows us to make a very limited use of formal logic, in constrast with most usual nonstandard analysisintroductions.

Apart from being framed in a non-Archimedean setting, ultrafunctions have other peculiar propertiesthat will be introduced and used in the following:

• every ultrafunction can be splitted uniquely (in a sense that will be precised in Section 3.9) as thesum of a classical function and a purely non-Archimedean part;• ultrafunctions extend distributions, in the sense that every distribution can be identified with an

ultrafunction; in particular, this allows to perform nonlinear operations with distribution;• although being generalized functions, ultrafunctions shares many properties of C1 functions, like

e.g. Gauss’ divergence theorem.

Our goal is to introduce all the aforementioned properties of ultrafunctions so to be able to explainhow they can be used to solve certain classical problems that do not have classical solutions; in particular,we will concentrate on singular problems arising in calculus of variations and in relevant applications(see, e.g., [18] and references therein for other approaches to these problems based on different notions ofgeneralized functions).

The paper is organized as follows: in Section 2 we introduce the notion of Λ-limit, and we explain howto use it to construct all the non-Archimedean tools that are needed in the rest of the paper, in particularhow to construct the non-Archimedean field extension R∗ of R and what the notion of ”hyperfinite” means.In Section 3 we define ultrafunctions, and we explain how to extend derivatives and integrals to them. Allthe properties of ultrafunctions needed later on are introduced in this section: we show how to split anultrafunction as the sum of a standard and a purely non-Archimedean part, how to extend Gauss’ divergencetheorem and how to identify distributions with certain ultrafunctions. In Section 4 we present the mainresults of the paper, namely we show that a very large class of classical problems admits a generalizedultrafunction solutions. We study the main properties of these generalized solutions, concentrating inparticular on the relationships between ultrafunction solutions and classical minimizing sequences forvariational problems. Finally, in Section 5 we present two examples of applications of our methods: thefirst is the study of a variational problem related with the sign-perturbation of potentials, the second is asingular variation problem related with sign-changing boundary conditions.

The first part of this paper contains some overlap with other papers on ultrafunctions, but this fact isnecessary to make it self-contained and to make the reader comfortable with it.

1.1. Notations. If X is a set and Ω is a subset of RN , then

• P (X) denotes the power set of X and Pfin (X) denotes the family of finite subsets of X;• F (X,Y ) denotes the set of all functions from X to Y and F (Ω) = F (Ω,R) .• C (Ω) denotes the set of continuous functions defined on Ω ⊂ RN ;• Ck (Ω) denotes the set of functions defined on Ω ⊂ RN which have continuous derivatives up to the

order k (sometimes we will use the notation C 0(Ω) instead of C (Ω) );• Hk,p (Ω) denotes the usual Sobolev space of functions defined on Ω ⊂ RN• if W (Ω) is any function space, then Wc (Ω) will denote de function space of functions in W (Ω)

having compact support;• C0 (Ω ∪ Ξ) , Ξ ⊆ ∂Ω, denotes the set of continuous functions in C (Ω ∪ Ξ) which vanish for x ∈ Ξ• D (Ω) denotes the set of the infinitely differentiable functions with compact support defined on

Ω ⊂ RN ; D′ (Ω) denotes the topological dual of D (Ω), namely the set of distributions on Ω;

1We refer to Keisler [17] for a very clear exposition of nonstandard analysis.

GENERALIZED SOLUTIONS OF PDES AND APPLICATIONS 3

• if A ⊂ X is a set, then χA denotes the characteristic function of A;• supp(f) = supp∗(f) where supp is the usual notion of support of a function or a distribution;

• mon(x) = y ∈(RN)∗

: x ∼ y where x ∼ y means that x− y is infinitesimal;• ∀a.e. x ∈ X means ”for almost every x ∈ X”;• if a, b ∈ R∗, then

– [a, b]R∗ = x ∈ R∗ : a ≤ x ≤ b;– (a, b)R∗ = x ∈ R∗ : a < x < b;

• if W is a generic function space, its topological dual will be denoted by W ′ and the pairing by〈·, ·〉W ;• if E is any set, then |E| will denote its cardinality.

2. Λ-theory

In this section we present the basic notions of Non Archimedean Mathematics (sometimes abbreviatedas NAM) and of Nonstandard Analysis (sometimes abbreviated as NSA) following a method inspired by [3](see also [1] and [4]).

2.1. Non Archimedean Fields. Here, we recall the basic definitions and facts regarding non-Archimedeanfields. In the following, K will denote a totally ordered infinite field. We recall that such a field contains (acopy of) the rational numbers. Its elements will be called numbers.

Definition 2.1. Let K be an ordered field. Let ξ ∈ K. We say that:

• ξ is infinitesimal if, for all positive n ∈ N, |ξ| < 1n ;

• ξ is finite if there exists n ∈ N such as |ξ| < n;• ξ is infinite if, for all n ∈ N, |ξ| > n (equivalently, if ξ is not finite).

Definition 2.2. An ordered field K is called Non-Archimedean if it contains an infinitesimal ξ 6= 0.

It is easily seen that infinitesimal numbers are actually finite, that the inverse of an infinite number is anonzero infinitesimal number, and that the inverse of a nonzero infinitesimal number is infinite.

Definition 2.3. A superreal field is an ordered field K that properly extends R.

It is easy to show, due to the completeness of R, that there are nonzero infinitesimal numbers and infinitenumbers in any superreal field. Infinitesimal numbers can be used to formalize a new notion of closeness,according to the following

Definition 2.4. We say that two numbers ξ, ζ ∈ K are infinitely close if ξ − ζ is infinitesimal. In thiscase, we write ξ ∼ ζ.

Clearly, the relation ∼ of infinite closeness is an equivalence relation and we have the following

Theorem 2.5. If K is a totally ordered superreal field, every finite number ξ ∈ K is infinitely close to aunique real number r ∼ ξ, called the standard part of ξ.

Given a finite number ξ, we denote its standard part by st(ξ), and we put st(ξ) = ±∞ if ξ ∈ K is apositive (negative) infinite number. In Definition 2.16 we will see how the notion of standard part can begeneralized to any Hausdorff topological space.

Definition 2.6. Let K be a superreal field, and ξ ∈ K a number. The monad of ξ is the set of all numbersthat are infinitely close to it

mon(ξ) := ζ ∈ K : ξ ∼ ζ.

2.2. The Λ-limit. Let U be an infinite set of cardinality bigger that the continuum and let L = Pfin(U)be the family of finite subsets of U .

Notice that (L,⊆) is a directed set. We add to L a point at infinity Λ /∈ L, and we define the followingfamily of neighbourhoods of Λ :

Λ ∪Q | Q ∈ U,where U is a fine ultrafilter on L, namely a filter such that


• for every A,B ⊆ L, if A ∪B = L then A ∈ U or B ∈ U ;• for every λ ∈ L the set Q(λ) := µ ∈ L | λ v µ ∈ U .

We will refer to the elements of U as qualified sets. A function ϕ : L→ E defined on a directed set E iscalled net (with values in E). If ϕ(λ) is a real net, we have that

limλ→Λ

ϕ(λ) = L

if and only if

∀ε > 0, ∃Q∈U : ∀λ∈Q, |ϕ(λ)− L| < ε.

As usual, if a property P (λ) is satisfied by any λ in a neighbourhood of Λ we will say that it is eventuallysatisfied.

Proposition 2.7. If the net ϕ(λ) takes values in a compact set K, then it is a converging net.

Proof. Suppose that the net ϕ(λ) has a converging subnet to L ∈ R. We fix ε > 0 arbitrarily and we haveto prove that Qε ∈ U where

Qε = λ ∈ L | |ϕ(λ)− L| < ε .We argue indirectly and we assume that

Qε /∈ U .Then, by the definition of ultrafilter, N = L\Qε ∈ U and hence

∀λ ∈ N, |ϕ(λ)− L| ≥ ε.

This contradict the fact that ϕλ has a subnet which converges to L.

Proposition 2.8. Assume that ϕ : L→ E where E is a first countable topological space; then if

limλ→Λ

ϕ(λ) = x0,

there exists a sequence λn in L such that

limn→∞

ϕ(λn) = x0

We refer to the sequence ϕn := ϕ(λn) as a subnet of ϕ(λ).

Proof. Let An | n ∈ N be a countable basis of open neighbourhoods of x0. For every n ∈ N the set

In := λ ∈ L | ϕ(λ) ∈ An

is qualified. Hence Jn :=⋂j≤n Ij 6= ∅. Let λn ∈ Jn. Then the sequence λnn∈N has trivially the desired

property: for every n ∈ N, for every m ≥ n we have that ϕ (λm) ∈ An.

Example 2.9. Let ϕ : L→ V be a net with values in a bounded subset of a reflexive Banach space equippedwith the weak topology; then

v := limλ→Λ

ϕ(λ),

is uniquely defined and there exists a sequence n 7→ ϕ(λn) which converges to v.

Definition 2.10. The set of the hyperreal numbers R∗ ⊃ R is a set equipped with a topology τ such that

• every net ϕ : L → R has a unique limit in R∗ if L and R∗ are equipped with the Λ and the τtopology respectively;• R∗ is the closure of R with respect to the topology τ ;• τ is the coarsest topology which satisfies the first property.


The existence of such R∗ is a well known fact in NSA. The limit ξ ∈ R∗ of a net ϕ : L→ R with respectto the τ topology, following [1], is called the Λ-limit of ϕ and the following notation will be used:

(2.1) ξ = limλ↑Λ

ϕ(λ);

namely, we shall use the up-arrow “↑” to remind that the target space is equipped with the topology τ .Given

ξ := limλ↑Λ

ϕ(λ), η := limλ↑Λ

ψ(λ),

we set

(2.2) ξ + η := limλ↑Λ

(ϕ(λ) + ψ(λ)) ,

and

(2.3) ξ · η := limλ↑Λ

(ϕ(λ) · ψ(λ)) .

Then the following well known theorem holds:

Theorem 2.11. The definitions (2.2) and (2.3) are well posed and R∗, equipped with these operations, isa non-Archimedean field.

Remark 2.12. We observe that the field of hyperreal numbers is defined as a sort of completion of realnumbers. In fact R∗ is isomorphic to the ultrapower

RL/I

where

I = ϕ : L→ R |ϕ(λ) = 0 eventuallyThe isomorphism resembles the classical one between real numbers and equivalence classes of Cauchysequences: this method is surely known to the reader for the construction of the real numbers starting fromthe rationals.

2.3. Natural extension of sets and functions. To develop applications, we need to extend the notionof Λ-limit to sets and functions (but also to differential and integral operators). This will allow to enlargethe notions of variational problem and of variational solution.

Λ-limits of bounded nets of mathematical objects in V∞(R) can be defined by induction (a net ϕ : L→V∞(R) is called bounded if there exists n ∈ N such that ∀λ ∈ L, ϕ(λ) ∈ Vn(R)). To do this, consider a net

(2.4) ϕ : L→ Vn(R).

Definition 2.13. For n = 0, limλ↑Λ ϕ(λ) is defined by (2.1); so by induction we may assume that the limitis defined for n− 1 and we define it for the net (2.4) as follows:

limλ↑Λ

ϕ(λ) =

limλ↑Λ

ψ(λ) | ψ : L→ Vn−1(R), ∀λ ∈ L, ψ(λ) ∈ ϕ(λ)

.

A mathematical entity (number, set, function or relation) which is the Λ-limit of a net is called internal.

Definition 2.14. If ∀λ ∈ L, Eλ = E ∈ V∞(R), we set limλ↑Λ Eλ = E∗, namely

E∗ :=

limλ↑Λ

ψ(λ) | ψ(λ) ∈ E.

E∗ is called the natural extension of E.

Notice that, while the Λ-limit of a constant sequence of numbers gives this number itself, a constantsequence of sets gives a larger set, namely E∗. In general, the inclusion E ⊆ E∗ is proper.

Given any set E, we can associate to it two sets: its natural extension E∗ and the set Eσ, where

(2.5) Eσ := X∗ | X ∈ E .Clearly Eσ is a copy of E; however it might be different as set since, in general, X∗ 6= X.


Remark 2.15. If ϕ : L→ X is a net with value in a topological space we have the usual limit

limλ→Λ

ϕ(λ),

which, by Proposition 2.7, always exists in the Alexandrov compactification X ∪ ∞. Moreover we havethe Λ-limit, that always exists and it is en element of X∗. In addition, the Λ-limit of a net is in Xσ if andonly if ϕ is eventually constant. If X = R and both limits exist, then

(2.6) limλ→Λ

ϕ(λ) = st

(limλ↑Λ

ϕ(λ)

).

The above equation suggests the following definition.

Definition 2.16. If X is topological space equipped with a Hausdorff topology, and ξ ∈ X∗ we set

StX (ξ) = limλ→Λ

ϕ(λ),

if there is a net ϕ : L→ X converging in the topology of X and such that

ξ = limλ↑Λ

ϕ(λ),

and StX (ξ) =∞ otherwise.

By the above definition we have that

limλ→Λ

ϕ(λ) = StX

(limλ↑Λ

ϕ(λ)

).

Definition 2.17. Let

fλ : Eλ → R, λ ∈ L,

be a net of functions. We define a function

f :

(limλ↑Λ

Eλ

)→ R∗

as follows: for every ξ ∈ (limλ↑Λ Eλ) we set

f (ξ) := limλ↑Λ

fλ (ψ(λ)) ,

where ψ(λ) is a net of numbers such that

ψ(λ) ∈ Eλ and limλ↑Λ

ψ(λ) = ξ.

A function which is a Λ-limit is called internal. In particular if, ∀λ ∈ L,

fλ = f, f : E → R,

we set

f∗ = limλ↑Λ

fλ.

f∗ : E∗ → R∗ is called the natural extension of f. If we identify f with its graph, then f∗ is the graphof its natural extension.


2.4. Hyperfinite sets and hyperfinite sums.

Definition 2.18. An internal set is called hyperfinite if it is the Λ-limit of a net ϕ : L→ F where F is afamily of finite sets.

For example, if E ∈ V∞(R), the set

E = limλ↑Λ

(λ ∩ E)

is hyperfinite. Notice that

Eσ ⊂ E ⊂ E∗

so, we can say that every standard set is contained in a hyperfinite set.It is possible to add the elements of a hyperfinite set of numbers (or vectors) as follows: let

A := limλ↑Λ

Aλ,

be a hyperfinite set of numbers (or vectors); then the hyperfinite sum of the elements of A is defined in thefollowing way: ∑

a∈Aa = lim

λ↑Λ

∑a∈Aλ

a.

In particular, if Aλ =a1(λ), ..., aβ(λ)(λ)

with β(λ) ∈ N, then setting

β = limλ↑Λ

β(λ) ∈ N∗,

we use the notationβ∑j=1

aj = limλ↑Λ

β(λ)∑j=1

aj(λ).

3. Ultrafunctions

3.1. Definition of ultrafunctions. We start by introducing the notion of hyperfinite grid:

Definition 3.1. A hyperfinite set Γ such that RN ⊂ Γ ⊂(RN)∗

is called hyperfinite grid.

From now on we assume that Γ has been fixed once forever. Notice that, by definition, RN ⊆ Γ, and thefollowing two simple (but useful) properties of Γ can be easily proven via Λ-limits:

• for every x ∈ RN there exists y ∈ Γ ∩mon(x) so that x 6= r;• there exists a hyperreal number ρ ∼ 0, ρ > 0 such that d(x, y) ≥ ρ for every x, y ∈ Γ, x 6= y.

Definition 3.2. A space of grid functions is a family G(RN ) of internal functions

u : Γ→ R∗

defined on a hyperfinite grid Γ. If E ⊂ RN , then G(E) will denote the restriction of the grid functions tothe set E∗ ∩ Γ.

Let E be any set in RN . To every internal function u ∈ F(E)∗, it is possible to associate a grid functionby the “restriction” map

(3.1) : F(E)∗ → G(E)

defined as follows:

∀x ∈ E∗ ∩ Γ, u(x) := u∗(x);

moreover, if f ∈ F(E), for short, we use the notation

(3.2) f(x) := (f∗)

(x).

So every function f ∈ F(E), can be uniquely extended to a grid function f ∈ G(E).In many problem we have to deal with functions defined almost everywhere in Ω, such as 1/|x|. Thus it

is useful to give a “rule” which allows to define a grid function for every x ∈ Γ:


Definition 3.3. If a function f is defined on a set E ⊂ RN , we put

f(x) =∑

a∈Γ∩E∗

f∗(a)σa(x),

where ∀a ∈ Γ the grid fuction σa is defined as follows: σa(x) := δax.

If E ⊂ RN is a measurable set, we define the “density function” of E as follows:

(3.3) θE(x) = st

(m(Bη(x) ∩ E∗)m(Bη(x))

),

where η is a fixed infinitesimal and m is the Lebesgue measure. Clearly θE(x) is a function whose value is 1in int(E) and 0 in RN \E; moreover, it is easy to prove that θE(x) is a measurable function and we havethat ˆ

θE(x)dx = m(E)

whenever m(E) <∞; also, if E is a bounded open set with smooth boundary, we have that θE(x) = 1/2for every x ∈ ∂E.

Now let V (RN ) be a vector space such that D(RN ) ⊂ V (RN ) ⊂ L 1(RN ).

Definition 3.4. A space of ultrafunctions V (RN ) modelled over the space V (RN ) is a space of gridfunctions such that there exists a vector space VΛ(RN ) ⊂ V ∗(RN ) such that the map2

: VΛ(RN )→ V ∗(RN )

is a R∗-linear isomophism. From now on, we assume that V (RN ) satisfies the following assumption: if Ωis a bounded open set such that mN−1(∂Ω) <∞ and f ∈ C0(RN ), then

fθΩ ∈ V (RN ).

Next we want to equip V (RN ) with the two main operations of calculus: the integral and the derivative.

Definition 3.5. The pointwise integral“: V (RN )→ R∗

is a linear functional which satisfies the following properties:

(1) ∀u ∈ VΛ(RN )

(3.4)

“u(x) dx =

ˆu(x)dx;

(2) there exists a ultrafunction d : Γ→ R∗ such that ∀x ∈ Γ, d(x) > 0 and ∀u ∈ V (RN ),“u(x) dx =

∑a∈Γ

u(a)d(a).

If E ⊂ RN is any set, we use the obvious notation“E

u(x) dx :=∑

a∈Γ∩E∗

u(a)d(a)

Few words to discuss the above definition. Point 2 says that the poinwise integral is nothing else but ahyperfinite sum. Since d(x) > 0 every non-null positive ultrafunction has a strictly positive integral. Inparticular if we denote by σa(x) the ultrafunctions whose value is 1 for x = a and 0 otherwise, we have that“

σa(x) dx = d(a).

2V ∗(E) is a shorthand notation for [V (E)]∗


The pointwise integral allows us to define the following scalar product:

(3.5)

“u(x)v(x)dx =

∑a∈Γ

u(a)v(a)d(a).

From now on, the norm of an ultrafunction will be defined as

‖u‖ =

(“|u(x)|2 dx

) 12

.

Now let us examine the point 1 of the above definition. If we take f ∈ C0comp(RN ), we have that f∗ ∈ VΛ(RN )

and hence “f(x) dx =

ˆf(x)dx.

Thus the pointwise integral is an extension of the Riemann Integral defined on C0comp(RN ). However, if we

take a bounded open set Ω such that m(∂Ω) = 0, then we have thatˆΩ

f(x)dx =

ˆΩ

f(x)dx;

however, the pointwise integral cannot have this property; in fact“Ω

f(x)dx−“

Ω

f(x)dx =

“∂Ω

f(x)dx > 0

since ∂Ω 6= ∅. In particular, if Ω is a bounded open set with smooth boundary and f ∈ C0(RN ), then“Ω

f(x) dx =

“f(x)χΩ(x) dx

=

“f(x)θΩ(x) dx− 1

2

“f(x)χ∂Ω(x) dx

=

ˆΩ

f(x) dx− 1

2

“∂Ω

f(x) dx,

and similarly “Ω

f(x) dx =

ˆΩ

f(x) dx+1

2

“∂Ω

f(x) dx;

of course, the term 12

›f(x)χ∂E(x) dx is an infinitesimal number and it is relevant only in some particular

problems.

Definition 3.6. The ultrafunction derivative

Di : V (RN )→ V (RN )

is a linear operator which satisfies the following properties:

(1) ∀f ∈ C1(RN ), and ∀x ∈ (RN )∗, x finite,

(3.6) Dif(x) = ∂if

∗(x);

(2) ∀u, v ∈ V (RN ), “Diuv dx = −

“uDiv dx;

(3) if Ω is a bounded open set with smooth boundary, then ∀v ∈ V ,“DiθΩv dx = −

ˆ ∗∂Ω

v (ei · nE) dS

where nE is the unit outer normal, dS is the (n− 1)-dimensional measure and (e1, ...., eN ) is thecanonical basis of RN .

(4) the support3 of Diσa is contained in mon(a) ∩ Γ.

3If u is an ultrafunction the support of u is the set x ∈ Γ | u(x) 6= 0.


Let us comment the above definition. Point (1) implies that ∀f ∈ C1(RN ), and ∀x ∈ RN ,

(3.7) Dif(x) = ∂if(x);

namely the ultrafunction derivative concides with the usual partial derivative whenever f ∈ C1(RN ). Themeaning of point (2) is clear; we remark that this point is very important in comparing ultrafunctions withdistributions. Point (3) says that DiθΩ is an ultrafunction whose support is contained in ∂Ω ∩ Γ; it can beconsidered as a signed measure concentrated on ∂Ω. Point (4) says that the ultrafunction derivative, as wellas the usual derivative or the distributional derivative, is a local operator, namely if u is an ultrafunctionwhose support is contained in a compact set K with K ⊂ Ω, then the support of Diu is contained in Ω∗.Moreover, property (4) implies that the ultrafunction derivative is well defined in V (Ω) for any open set Ωby the following formula:

Diu(x) =∑

a∈Γ∩Ω∗

u(a)Diσa(x).

Remark 3.7. If u ∈ V (Ω) and u is an ultrafunction in V (Ω) such that ∀x ∈ Ω, u(x) = u(x), then, bypoint (3), ∀x ∈ Ω∗ such that mon(x) ⊂ Ω∗ we have that

Diu(x) = Diu(x);

however, this property fails for some x ∼ ∂Ω∗. In fact the support of Diσa is contained in mon(a) ∩ Γ, butnot in a.

Theorem 3.8. There exists a ultrafuctions space V (RN ) which admits a pointwise integral and a ultra-function derivative as in Def. 3.5 and 3.6.

Proof. In [2] there is a construction of a space VΛ(RN ) which satisfies the desired properties. The conclusionfollows taking

V (RN ) = u : u ∈ VΛ .

3.2. The splitting of an ultrafunction. In many applications, it is useful to split an ultrafunction u ina part w which is the canonical extension of a standard function w and a part ψ which is not directlyrelated to any classical object. If u ∈ V (Ω), we set

S = x ∈ Ω | u(x) is infinite

and

w(x) =

st(u(x)) if x ∈ Ω \ S;

0 if x ∈ S.We will refer to S as to the singular set of the ultrafunction u.

Definition 3.9. For every ultrafunction u consider the splitting

u = w + ψ

where

• w = w|Ω\S and w, which is defined by Def. 3.3, is called the functional part of u;• ψ := u− w is called the singular part of u.

Notice that w, the functional part of u, may assume infinite values for some x ∈ Ω∗ \ S∗, but they aredetermined by the values of w which is a standard function defined on Ω.

Example 3.10. Take ε ∼ 0, and

u(x) = log(x2 + ε2).

In this case

• w(x) =

log(x2) = 2 log(|x|) if x 6= 0

0 if x = 0


• ψ(x) =

log(x2 + ε2)− log(x2) = log

(1 + ε2

x2

)if x 6= 0

log(ε2) if x = 0

• S := 0 .

We conclude this section with the following trivial proposition which, nevertheless, is very useful inapplications:

Proposition 3.11. Take a Banach space W such that D(Ω) ⊂W ⊂ L1(Ω). Assume that un ⊆ V (Ω) isa sequence which converges weakly in W and pointwise to a function w; then, if we set

u :=

((limλ↑Λ

u|λ|

)),

we have that

u = w + ψ,

where

∀v ∈W,“ψv dx ∼ 0;

moreover, if

limn→∞

‖un − w‖W = 0

then ‖ψ‖W ∼ 0.

Proof. As a consequence of the pointwise convergence of un to w we have that ∀a ∈ Γu(a) ∼ w(a). Inparticular, ∀a ∈ Γψ(a) ∼ 0. As Γ is hyperfinite, the set |ψ(a)| | a ∈ Γ has a max η ∼ 0. Hence for everyv ∈W we have ∣∣∣∣“ ψv dx

∣∣∣∣ ≤ “ |ψ| |v| dx ≤ η “ |v| dx ∼ η ˆ |v|dx ∼ 0,

as η ∼ 0 and´|v|dx ∈ R. For the second statement let us notice that

‖ψ‖W = ‖u− w‖W =

(limλ↑Λ

∥∥u|λ| − w∥∥W) ∼ 0

as limn→∞ ‖un − w‖W = 0.

An immediate consequence of Prop. 3.11 is the following:

Corollary 3.12. If w ∈ L1(Ω) then “w(x)dx ∼

ˆw(x)dx.

Proof. Since V (Ω) is dense in L1(Ω) there is a sequence un ∈ V (Ω) which converges strongly to w in L1(Ω).Now set

u :=

(limλ↑Λ

u|λ|

).

By Prop. 3.11, we have that

u = w + ψ

with ‖ψ‖L1 ∼ 0. Then “u(x)dx ∼

“w(x)dx.

On the other hand, since u ∈ V (Ω), by Def. 3.5.(1),“u(x)dx =

ˆ ∗u(x)dx = lim

λ↑Λ

ˆu|λ|dx

∼ limn→Λ

ˆΩ

undx =

ˆw(x)dx.


3.3. The Gauss divergence theorem. First of all, we fix the notation for the main differential operators:

• ∇ = (∂1, ..., ∂N ) will denote the usual gradient of standard functions;• ∇∗ = (∂∗1 , ..., ∂

∗N ) will denote the natural extension of internal functions;

• D = (D1, ..., DN ) will denote the canonical extension of the gradient in the sense of ultrafunctions.

Next let us consider the divergence:

• ∇·φ = ∂1φ1 + ...+∂NφN will denote the usual divergence of standard vector fields φ ∈[C1(RN )

]N;

• ∇∗ · φ = ∂∗1φ1 + ...+ ∂∗NφN will denote the divergence of internal vector fields φ ∈[C1(RN )∗

]N;

• D ·φ = D1φ1 + ...+DNφN will denote the divergence of vector valued ultrafuction φ ∈[V (RN )∗

]N.

And finally, we can define the Laplace operator of an ultrafunction u ∈ V (Ω) as the only ultrafunction4u ∈ V (Ω) such that

∀v ∈ V 0 (Ω),

“4uvdx = −

“Du ·Dv dx,

where

V 0 (Ω) :=v ∈ V (Ω) | ∀x ∈ ∂Ω ∩ Γ, v(x) = 0

.

By definition 3.6.(3), for any bounded open set Ω with smooth boundary,“DiθΩv dx = −

ˆ ∗∂Ω

v (ei · nE) dS,

and by definition 3.6.(2) “DiθΩv dx = −

“Div θΩdx

so that “Div θΩdx =

ˆ ∗∂Ω

v (ei · nΩ) dS.

Now, if we take a vector field φ = (v1, ..., vN ) ∈[V (RN )

]N, by the above identity, we get

(3.8)

“D · φ θΩ dx =

ˆ ∗∂Ω

φ · nΩ dS.

Now, if φ ∈ C1, by definition 3.6.(1), we get the divergence Gauss theorem:ˆΩ

∇ · φdx =

ˆ∂Ω

φ · nE dS.

Then, (3.8) is a generalization of the Gauss’ theorem which makes sense for any bounded open set Ω withsmooth boundary and every vectorial ultrafunction φ. Next, we want to generalize Gauss’ theorem toany subset of A ⊂ RN . It is well known that, for any bounded open set Ω with smooth boundary, thedistributional derivative ∇θΩ is a vector valued Radon measure and we have that

〈|∇θΩ|, 1〉 = mN−1(∂Ω)

Then, the following definition is a natural generalization:

Definition 3.13. If A is a measurable subset of RN , we set

mN−1(∂Ω) :=

“|DθA| dx

and ∀v ∈ V (RN ),

(3.9)

“∂A

v(x) dS :=

“v(x) |DθA| dx.


Remark 3.14. Notice that “∂A

v(x) dS 6=“∂A

v(x) dx.

In fact the left hand term has been defined as follows:“∂A

v(x) dS =∑x∈Γ

v(x) |DθA(x)| d(x),

while the right hand term is “∂A

v(x) dx =∑

x∈Γ∩∂A∗

v(x) d(x);

in particular if ∂A is smooth and v(x) is bounded,›∂Av(x) dx is an infinitesimal number.

Theorem 3.15. If A is an arbitrary measurable subset of RN , we have that

(3.10)

“D · φ θA dx =

“∂A

φ · nA(x) dS,

where

nA(x) =

− DθA(x)|DθA(x)| if DθA(x) 6= 0;

0 if DθA(x) = 0.

Proof. By Definition 3.6.(3)

“D · φ θAdx = −

“φ ·DθAdx,

then, using the definition of nA(x) and (3.9), the above formula can be written as follows:“D · φ θAdx =

“φ · nA |DθA| dx =

“∂A

φ · nA dS.

3.4. Ultrafunctions and distributions. One of the most important properties of the ultrafunctions isthat they can be seen (in some sense that we will make precise in this section) as generalizations of thedistributions.

Definition 3.16. The space of generalized distribution on Ω is defined as follows:

D ′G(Ω) = V (Ω)/N,

where

N =

τ ∈ V (Ω) | ∀ϕ ∈ D(Ω),

ˆτϕ dx ∼ 0

.

The equivalence class of u in V (Ω) will be denoted by

[u]D .

Definition 3.17. Let [u]D be a generalized distribution. We say that [u]D is a bounded generalizeddistribution if ∀ϕ ∈ D(Ω),

´uϕ∗ dx is finite. We will denote by D ′GB(Ω) the set of bounded generalized

distributions.

We have the following result.

Theorem 3.18. There is a linear isomorphism

Φ : D ′GB(Ω)→ D ′(Ω)

such that, for every [u] ∈ D ′GB(Ω) and for every ϕ ∈ D(Ω)

〈Φ ([u]D) , ϕ〉D(Ω)= st

(“uϕ∗ dx

).


Proof. For the proof see e.g. [7].

From now on we will identify the spaces D ′GB(Ω) and D ′(Ω); so, we will identify [u]D with Φ ([u]D) andwe will write [u]D ∈ D ′(Ω) and

〈[u]D , ϕ〉D(Ω):= 〈Φ[u]D , ϕ〉 = st

(“uϕ∗ dx

).

If f ∈ C0comp(Ω) and f∗ ∈ [u]D , then ∀ϕ ∈ D(Ω),

〈[u]D , ϕ〉D(Ω)= st

(ˆ ∗uϕ∗ dx

)= st

(ˆ ∗f∗ϕ∗dx

)=

ˆfϕ dx.

Remark 3.19. The set V (Ω) is an algebra which extends the algebra of continuous functions C0(RN ).If we identify a tempered distribution4 T = ∂mf with the ultrafunction Dmf, we have that the set oftempered distributions S ′ is contained in V (Ω). However the Schwartz impossibility theorem is notviolated as (V (Ω),+, · , D) is not a differential algebra since the Leibnitz rule does not hold for some pairsof ultrafunctions. See also [7].

4. Properties of ultrafunction solutions

The problems that we want to study with ultrafunctions have the following form: minimize a givenfunctional J on V (Ω) subjected to certain restrictions (e.g., some boundary constrictions, or a minimizationon a proper vector subspace of V (Ω)). This kind of problems can be studied in ultrafunctions theoryby means of a modification of the Faedo-Galerkin method, based on standard approximations by finitedimensional spaces. The following is a (maybe even too) general formulation of this idea.

Theorem 4.1. Let W (Ω) 6= ∅ be a vector subspace of V (Ω). Let

F = f : V (Ω)→ R | ∀E finite dimensional vector subspace of W (Ω) ∃u ∈ E f(u) = minv∈E

f(v).

Then every F ∈ F∗ has a minimizer in WΛ(Ω).

Proof. Let F = limλ↑Λ fλ, with fλ ∈ F for every λ ∈ L. By hypothesis, for every λ ∈ L there existsuλ ∈Wλ := Span(W ∩λ) that minimizes fλ on Wλ. Then u = limλ↑Λ uλ minimizes F on limλ↑ΛWλ = WΛ

as, if v = limλ↑Λ vλ ∈Wλ(Ω), then for every λ ∈ L we have that fλ (vλ) ≤ fλ (uλ), hence

F (v) = limλ↑Λ

fλ (vλ) ≤ limλ↑Λ

fλ (uλ) = F (u).

For applications, the following particular case of Theorem 4.1 is particularly relevant:

Corollary 4.2. Let f(ξ, u, x) be cohercive in ξ on every finite dimensional subspace of V (Ω) and for everyx ∈ Ω. Let F (u) :=

›f(∇u, u, x)dx. Then F has a min on VΛ.

Proof. Just notice that F ∈ F , in the notations of Theorem 4.1.

Theorem 4.1 provides a general existence results. However, such a general result poses two questions: thefirst is how wild such generalized solutions can be; the second is if this method produces new generalizedsolutions for problems that already have classical ones.

The answer to these questions depends on the problem that is studied. However, regarding the secondquestion we have the following result, which strengthens Theorem 4.1:

Theorem 4.3. Let F : VΛ(Ω)→ R∗, F = limλ↑Λ Fλ. For every λ ∈ L let

Mλ :=

u ∈ Vλ(Ω) | Fλ(u) = min

v∈VλFλ(v)

.

4We recall that, by a well known theorem of Schwartz, any tempered distribution can be represented as ∂mf where m is a

multi-index and f is a continuous function.


Assume that limλ↑ΛMλ 6= ∅. Then

MΛ :=

u ∈ VΛ(Ω) | F (u) = min

v∈VΛ

F (v)

= limλ↑Λ

Mλ 6= ∅.

Proof. MΛ ⊆ limλ↑ΛMλ: Let v = limλ↑Λ vλ ∈ MΛ, and let u = limλ↑Λ uλ ∈ limλ↑ΛMλ. As F (v) ≤ F (u),there is a qualified set Q such that for every λ ∈ Q Fλ (vλ) ≤ F (uλ). But then vλ ∈Mλ for every λ ∈ Q,hence v = limλ↑Λ vλ ∈ limλ↑ΛMλ.MΛ ⊇ limλ↑ΛMλ: Let u = limλ↑Λ uλ ∈ limλ↑ΛMλ. Let v = limλ↑Λ vλ ∈ VΛ(Ω). Let

Q = λ ∈ L | uλ ∈Mλ.Then Q is qualified and, for every λ ∈ Q, Fλ (uλ) ≤ Fλ (vλ). Therefore F (u) ≤ F (v), and so u ∈MΛ.

The following easy consequences of Theorem 4.3 hold:

Corollary 4.4. In the same notations of Theorem 4.3, let us now assume that there exists k ∈ N suchthat |Mλ| ≤ k for every λ ∈ L. Then |MΛ| ≤ k.

Proof. This holds as the hypothesis on |Mλ| trivially entails that | limλ↑ΛMλ| ≤ k.

Corollary 4.5. In the same notations of Theorem 4.3, let us now assume that F = J∗, where J : V (Ω)→ R.Let

M :=

v ∈ V (Ω) | v = min

w∈V (Ω)J(w)

.

Assume that M 6= ∅. Then the following facts are equivalent:

(1) u is a minimizer of F : VΛ(Ω)→ R∗;(2) u ∈M∗ ∩ VΛ(Ω).

In particular, if u ∈M then u∗ minimizes F .

Proof. (1) ⇒ (2) Let u ∈ M . Let Q(u) := λ ∈ L | u ∈ λ. Then for every λ ∈ Q(u) v ∈ Mλ ⇔ J(v) =J(u)⇒ v ∈M , hence Mλ ⊆M for every λ ∈ Q(u), which is qualified, and so limλ↑ΛMλ ⊆M∗ ∩ VΛ, andwe conclude by Theorem 4.3.

(2)⇒ (1) By definition,u ∈M∗ ⇔ F (u) = min

v∈[V (Ω)]∗F (v),

hence if u ∈M∗ ∩ Vλ(Ω) it trivially holds that u minimizes F .

Corollary 4.6. In the same hypotheses and notations of Corollary 4.3, let us assume that M = u1, . . . , unis finite. Then v minimizes F in VΛ(Ω) if and only if there exists u ∈M such that u∗ = v.

Proof. Just remember that S = s∗|s ∈ S for every finite set S, and that Mσ = u∗ | u ∈M ⊆ V (Ω) ⊆VΛ(Ω).

In general, one might not have minima, but minimization sequences could still exist. In this case, wehave the following result (in which for every ρ ∈ R∗ we set stR(ρ) = −∞ if and only if ρ is a negativeinfinite number). Notice that in the following result we are not assuming the continuity of J with respectto any topology on V (Ω), in general.

Theorem 4.7. Let V (Ω) be a Banach space, let J : V (Ω)→ R and let infu∈V (Ω) J(u) = m ∈ R ∪ −∞.The following facts hold:

1. J∗(v) ≥ m for every v ∈ VΛ(Ω).2. There exists v ∈ VΛ(Ω) such that stR(J∗(v)) = m.3. If v ∈ VΛ(Ω) is a minimum of J∗ : VΛ(Ω)→ R∗ then J∗(v) ≥ stR (J∗(v)) = m.4. Let unn∈N be a minimizing sequence that converges to u ∈ V (Ω) in some topology τ . Then there

exists v ∈ VΛ(Ω) such that stτ (v) = u and J∗(v) ≥ stR (J∗(v)) = m. Moreover, if w + ψ is thecanonical splitting of v, then:• if τ is the topology of pointwise convergence, then w = v and w(x) = u(x) for every x ∈ Ω;• if τ is the topology of pointwise convergence a.e., then w = v and w(x) = u(x) a.e. in x ∈ Ω;


• if τ is the topology of weak convergence, then w(x) = u(x) for every x ∈ Ω and 〈ψ,ϕ∗〉∗ ∼ 0for every ϕ in the dual of V (Ω);

• if τ is the topology associated with a norm || · || and, moreover, unn converges pointwise tou, then w = u and ||ψ||∗ ∼ 0.

5. If all minimizing sequences of J converge to u ∈ V (Ω) in some topology τ and v is a minimum ofJ∗ : VΛ(Ω)→ R∗ then stτ (v) = u and J∗(v) ≥ stR(J∗(v)) = m.

Proof. (1) Let v = limλ↑Λ vλ. Since m = infu∈V (Ω) J(u) we have that for every λ ∈ Λ J (vλ) ≥ m, henceJ∗ (v) ≥ m.

(2) By (1) it sufficies to show that stR(J∗(v)) = m. Let unn∈N be a minimizing sequence for J . Forevery λ ∈ L let vλ := u|λ|. Let v := limλ↑Λ vλ. We claim that v is the desired ultrafunction.

To prove that stR(J∗(v)) = limn→+∞ J (un) = m we just have to observe that, by our definition of thenet vλλ, it follows that5

limn→+∞

J (un) = stR

(limλ↑Λ

J (vλ)

),

and we conclude as limλ↑Λ J (vλ) = J∗(v) by definition.(3) Let v = limλ↑Λ vλ, and let w ∈ VΛ(Ω) be such that stR (J∗(w)) = m. Then m ≤ J∗(v) by (1), whilst

stR (J∗(v)) ≤ stR (J∗(w)) = m. Hence st (J∗(v)) = m, as desired.(4) Let v be given as in point (2). Let us show that stV (Ω)(v) = u: let A ∈ τ be an open neighborhood

of u. As unn converges to u, there exists N > 0 such that for every m > N un ∈ A. Let µ ∈ L be suchthat |µ| > N . Then

∀λ ∈ Qµ := λ ∈ L | µ ⊆ λ vλ ∈ A,and as Qµ is qualified, this entails that v ∈ A∗. Since this holds for every A neighborhood of u, we

deduce that stτ (v) = u, as desired.Now let u = w + ψ be the splitting of u.If τ is the pointwise convergence, stτ (v)(x) = u(x) for every x ∈ Ω, hence by Definition 3.9 we have that

the singular set of u is empty and that w(x) = u(x) for every x ∈ Ω, as desired. A similar argument worksin the case of the pointwise convergence a.e.

If τ is the weak convergence topology, then stτ (v) = u means that 〈v, ϕ∗〉∗ ∼ 〈u, ϕ〉 for every ϕ in thedual of V (Ω). Now let S be the singular set of u. We claim that S = ∅. If not, let x ∈ S and let ϕ = δx.Then 〈v, ϕ∗〉∗ = v(x) is infinite, whilst 〈u, δx〉 = u(x) is finite, which is absurd. Henceforth for every x ∈ Ωwe have that ψ(x) = 0. But

〈u, ϕ〉 ∼ 〈v, ϕ∗〉∗ = 〈w + ψ,ϕ∗〉∗ = 〈w, ϕ∗〉∗ + 〈ψ,ϕ∗〉∗ = 〈w,ϕ〉+ 〈ψ,ϕ∗〉∗,hence stτ (ψ) = u− w. As ψ(x) = 0 for all x ∈ Ω, this means that u(x) = w(x) for every x ∈ Ω. Then

〈u, ϕ〉+ 〈ψ,ϕ∗〉∗ = 〈w,ϕ〉+ 〈ψ,ϕ∗〉∗ = 〈v, ϕ〉 ∼ 〈u, ϕ〉,and so 〈ψ,ϕ∗〉∗ ∼ 0.Finally, if τ is the strong convergence with respect to a norm || · || and unn converges pointwise to

u, then by what we proved above we have that v(x) ∼ u(x) ∀x ∈ Ω, hence u(x) ∼ w(x) for every x ∈ Ω,which means u = w as both u,w ∈ V (Ω). Then ||ψ|| = ||u− w|| = ||u− v||+ ||v − w|| ∼ 0.

(5) Let v = limλ↑Λ vλ. By point (2), the only claim to prove is that stτ (v) = u. We distinguish twocases:

Case 1: J∗(v) ∼ r ∈ R. As we noticed in point (2), it must be r = m. By contrast, let us assume thatstτ (v) 6= u. In this case, there exists an open neighborhood A of u such that the set

Q := λ ∈ L | vλ /∈ Ais qualified. For every n ∈ N, let

5A proof of this simple claim is given in Lemma 28 in [11].


Qn :=

λ ∈ L | |J(vλ)− r| < 1

n

∩Q.

Every Qn is qualified, hence nonempty. For every n ∈ N, let λn ∈ Qn. Finally, set un := vλn . Byconstruction, limn∈N J (un) = m. This means that unn∈N is a minimizing sequence, hence it converges tou in the topology τ , and this is absurd as, for every n ∈ N, by construction un /∈ A. Henceforth stτ (v) = u.

Case 2: J∗(v) ∼ −∞. As we noticed in the proof of point (2), in this case m = −∞. Let us assume thatstV (Ω)(v) 6= u. Then there exists an open neighborhood A of u such that the set

Q := λ ∈ L | vλ /∈ Ais qualified. For every n ∈ N, let

Qn = λ ∈ L | J(vλ) < −n ∩Qand let λn ∈ Qn. Finally, let un := vλn . Then J (un) < −n for every n ∈ N, hence unn∈N is a

minimizing sequence, and so it must converge to u. However, by construction un /∈ A for every n ∈ N,which is absurd.

Example 4.8. Let Ω = (0, 1), let

V (Ω) = u : Ω→ R | u is the restriction to Ω of a piecewise C1([0, 1]) functionand let J : V (Ω)→ R be the functional

J(u) :=

ˆΩ

u2(x)dx+

ˆΩ

((u′)

2 − 1)2

dx.

It is easily seen that infu∈V (Ω) J(u) = 0, and that the minimizing sequences of J converge pointwise and

strongly in the L2 norm to 0, but J(0) = 1.Let v ∈ VΛ(Ω) be the minimum of J∗ : VΛ(Ω). From points (4) and (5) of Theorem 4.7 we deduce that

0 < J∗(v) ∼ 0, that stV (Ω)(v) = 0 and that the canonical decomposition of v is v = 0 + ψ, with ψ = 0 for

every x ∈ Ω and´ ∗

Ω∗ ψ2dx ∼ 0. Moreover, as J∗(ψ) = 0, we also have that

´ ∗Ω∗

((ψ′)

2 − 1)2

dx ∼ 0.

5. Applications

5.1. Sign-perturbation of potentials. The first problem that we would like to tackle by means ofultrafunctions regards the sign-perturbation of potentials.

Let us start by recalling some results recently proved by L. Brasco and M. Squassina in [13] as arefinement and extension of some classical result by Brezis and Nirenberg [14].Let Ω be a bounded domain of RN with6 N > 2. Consider the minimization problem

(5.1) S(a) := infu∈D1,2

0 (Ω)

‖∇u‖2L2(Ω) +

ˆΩ

a |u|2 dx : ‖u‖L2∗ (Ω) = 1

,

where a ∈ LN/2(Ω) is given, 2∗ = 2N/(N − 2),

D1,20 (Ω) :=

u ∈ L2∗

(Ω) | ∇u ∈ L2(Ω), u = 0 on ∂Ω.

By Lagrange multipliers rule, minimizers of the previous problem (provided they exist) are constant signweak solutions of

(5.2)

−∆u+ a u = µ |u|2∗−2 u, in Ω,

u = 0, on ∂Ω,

with µ = S(a), namely ˆΩ

∇u · ∇ϕdx+

ˆΩ

a uϕdx = µ

ˆΩ

|u|2∗−2 uϕdx,

6In [13] the authors work more in general with a p ∈ (1, N), and consider also a fractional version of Problem 5.2; however,

in this paper we prefer to consider only the local case p = 2.


for every ϕ ∈ D1,20 (Ω).

The main result in [13] is the following Theorem, where the standard notations

a+ = maxa, 0, a− = max−a, 0, BR(x0) = x ∈ RN : |x− x0| < Rare used:

Theorem 5.1 (Brasco, Squassina). Let Ω ⊂ RN be an open bounded set. Then the following facts hold:

1. If a ≥ 0, then S(a) does not admit a solution.2. Let N > 4. Assume that there exist σ > 0, R > 0 and x0 ∈ Ω such that

a− ≥ σ, a. e. on BR(x0) ⊂ Ω.

Then S(a) admits a solution.3. Let 2 < N ≤ 4. For any x0 ∈ Ω, for any R > 0 s.t. BR(x0) ⊂ Ω there exists σ = σ(R,N) > 0 such

that ifa− ≥ σ, a. e. on BR(x0),

then S(a) admits a solution.

In [1], V. Benci studied in the ultrafunctions setting the following similar (simpler) problem: minimize

minu∈Mp

J(u),

where

J(u) =

ˆΩ

|∇u|2 dx

and

Mp =

u ∈ C2

0(Ω) :

ˆΩ

|u|p dx = 1

.

Here Ω is a bounded set in RN with smooth boundary, N ≥ 3 and p > 2. In the ultrafunctions settingintroduced in [1] (and with the notations of [1]), the problem takes the following form:

(5.3) minu∈Mp

J(u),

where

J(u) =

ˆ ∗Ω

|∇u|2 dx

and

Mp =

u ∈ V 2,0

B (Ω) |ˆ ∗

Ω

|u|p dx = 1

with V 2,0

B (Ω) = B[C2

0(Ω)].

For every p > 2, problem (5.3) has a ultrafunction solution up and, by setting mp = J(up), one can showthat

• (i) if 2 < p < 2∗, then mp = mp ∈ R+ and there is at least one standard minimizer up, namely

up ∈ C20(Ω);

• (ii) if p = 2∗ (and Ω 6= RN ), then m2∗ = m2∗ + ε where ε is a positive infinitesimal;• (iii) if p > 2∗, then mp = εp where εp is a positive infinitesimal.

Our goal is to show that a similar result can be obtained for Problem 5.2.In the present ultrafunctions setting, Problem 5.2 takes the following form: find

(5.4) S(a) := infu∈VΛ(Ω)

ˆ ∗(RN )∗

|∇u|2 dx+

ˆ ∗(RN )∗

a|u|2 dx : ‖u‖[L2∗ (RN )]∗ = 1

,

where a ∈ ∗[LN/2(Ω)

]is given, and VΛ(Ω) =

[D1,2

0 (Ω)]

Λ. With the above notations, we can prove the

following:


Theorem 5.2. Let Ω ⊂ RN be an open bounded set. Then the following facts hold:

(1) For every a ∈[LN/2(Ω)

]∗there exists u ∈ VΛ(Ω) that minimizes S(a).

(2) Let a ∈[LN/2(Ω)

]. If u ∈ C1 (Ω)∩ C0

(Ω)

is a minimizer of Problem 5.1 then u∗ is a minimizer of

S(a∗).

(3) If a = 0, then S(0) = S + ε, where

S := infu∈D0(RN )\0

‖∇u‖2L2

||u||2L2∗

and

ε =

0, if Ω = RN ,a strictly positive infinitesimal, if Ω 6= RN ;

moreover, if u is the minimizer in VΛ(Ω), then the functional part w of u is 0;(4) Let a ≥ 0 have an isolated minimum xm, and let u ∈ VΛ(Ω) be the minimum of Problem 5.4. If

u = w + ψ is the canonical splitting of u, then w = 0 and ψ concentrates in xm, in the sense thatfor every x /∈ mon(xm) ψ(x) ∼ 0. Moreover, 〈ψ,ϕ∗〉∗ ∼ 0 for every ϕ in the dual of V (Ω).

Proof. (1) This follows from Theorem 4.3, as the functional ‖∇u‖2L2(Ω) +´

Ωa |u|2 dx admits a minimum

on every finite dimensional subspace of VΛ.(2) This follows from Corollary 4.5.(3) In [13, Lemma 3.1] it was proved that, if we consider problem 5.1, we have that

S(0) = S

and S(0) is attained in D0(Ω) if and only if Ω = RN . Therefore if Ω = RN the results follows from point

(2). If Ω 6= RN , the fact that S(0) = S + ε follows from Theorem 4.7.(3). Moreover, all minimizingsequences unn converge weakly to 0 in H1, therefore they converges strongly in L2(Ω) and so theyconverge pointwise a.e., hence by Theorem 4.7.(5) we deduce that, in the splitting u = w + ψ, we havethat w = 0, namely the ultrafunction solution coincides with its singular part.

(4) We start by following the approach of [13]: we let U be a minimizer of

infu∈D1,2

0 (Ω)

[u]2D1,2

‖u‖2L2∗

and, for every ε > 0, let Uε(r) := ε2−N

2 U(rε

). Let δ > 0 be such that Bδ(xm) ⊆ Ω, and let uδ,ε be defined

as follows

uδ,ε =

Uε(r) if r ≤ δ,Uε(δ)

Uε(r)−Uε(δΘ)Uε(δ)−Uε(δΘ) if δ < r ≤ δΘ;

0 if r > δΘ;

where Θ is a constant given in Lemma 2.4 of [13]. Moreover, if F (u) := ‖∇u‖2L2(Ω) +´

Ωa |u|2 dx, for δ1, δ2

small enough we have that F (uδ1,ε) ≤ F (uδ2,ε). Then (uε,ε) is a minimizing net (for ε → 0), so we canuse Theorem 4.7.(4): as (uε,ε) converges pointwise to 0 we obtain that w = 0, whilst the definition of thenet ensure the concentration of ψ in xm. The last statement is again a direct consequence of Theorem4.7.(4).

Let us notice that the above Theorem shows a strong difference between the ultrafunctions and theclassical case: the existence of solutions in VΛ(Ω) is ensured independently of the sign of a whilst, asdiscussed in [13, Section 4], the conditions on a for the existence of solutions in the approach of L. Brascoand M. Squassina are essentially optimal. Of course, ultrafunction solutions might be very wild in general;their particular structure can be described in some cases, depending on a.

5.2. The singular variational problem.


5.2.1. Statement of the problem. Let W be a C1-function defined in R \ 0 such that

limt→0

W (t) = +∞

and

limt→±∞

W (t)

t2= 0.

We are interested in the singular problem (SP):

Naive formulation of Problem SP : find a continuous function

u : Ω→ R,which satisfies the equation:

(5.5) −∆u+W ′(u) = 0 in Ω

with the following boundary condition:

(5.6) u(x) = g(x) for x ∈ ∂Ω,

where Ω is an open set such that ∂Ω 6= ∅ and g ∈ L1(∂Ω) is a function different from 0 for every x whichchange sign; e.g. g(x) = ±1. Clearly, this problem does not have any solution in C1. This problem couldbe riformulated as a kind of free boundary problem in the following way:

Classical formulation of Problem SP : find two open sets Ω1 and Ω2 and two functions

ui : Ωi → R, i = 1, 2

such that all the following conditions are fulfilled:

Ω = Ω1 ∪ Ω2 ∪ Ξ where Ξ = Ω1 ∩ Ω2 ∩ Ω;

(5.7) −∆ui +W ′(ui) = 0 in Ωi, i = 1, 2;

ui(x) = g(x) for x ∈ ∂Ω ∩ ∂Ωi, i = 1, 2;

limx→Ξ

ui(x) = 0;

(5.8) Ξ is locally a minimal surface.

Condition (5.8) is natural, since formally equation (5.5) is the Euler-Lagrange equation relative to theenergy

(5.9) E(u) =1

2

ˆΩ

(|∇u|2 +W (u)

)dx

and the density of this energy diverges as x→ Ξ. In general this problem is quite involved since the set Ξcannot be a smooth surface and hence it is difficult to be characterized. However this problem becomesrelatively easy if formulated in the framework of ultrafuctions.

Let us recall that we the Laplace operator of a ultrafuction u ∈ V (Ω) is defined as the only ultrafunction∆u ∈ V (Ω) such that

∀v ∈ V 0 (Ω),

“Ω

∆uvdx = −“

Ω

Du ·Dv dx

whereV 0 (Ω) :=

v ∈ V (Ω) | ∀x ∈ ∂Ω ∩ Γ, v(x) = 0

.

Notice that, we can assert that ∆u(x) = D ·D(x) only in x ∂Ω∗.

Ultrafunction formulation of Problem SP7: find u ∈ V (Ω) such that

(5.10) u(x) 6= 0, ∀x ∈(Ω)∗ ∩ Γ,

7If u is a ultrafunction and W,W ′, etc. are functions, for short, we shall write W (u),W ′(u), etc instead of W ∗(u), (W ′)∗ (u),

etc.


(5.11) −∆u+W ′(u) = 0, for x ∈ Ω∗ ∩ Γ

(5.12) u(x) = g(x), for x ∈ (∂Ω)∗ ∩ Γ.

As we will see in the next section, the existence of this problem can be easily proven using variationalmethods.

5.2.2. The existence result. The easiest way to prove the existence of a ultrafunction solution of problemSP is achieved exploiting the variational structure of equation (5.11). Let us consider the extension

(5.13) E(u) =

“Ω

(1

2|Du|2 +W (u)

)dx

of the functional (5.9) to the space

V g (Ω) :=u ∈ V (Ω) | ∀x ∈ (∂Ω)

∗ ∩ Γ, u(x) = g(x).

Remark 5.3. We remark that the integration is taken over Ω∗, but u is defined in Ω∗. This is important, in

fact for some x ∈ Ω∗, x ∼ ∂Ω∗, the value of Du(x) depends on the value of u in some point y ∈ ∂Ω∗, y ∼ x.This is a remarkable difference between the usual derivative and the ultrafunction derivative.

Lemma 5.4. Equation (5.11) is the Euler-Lagrange equation of the functional (5.13).

Proof. We use the expression of›

as given in Definition 3.6. As

E(u) =

“Ω

(1

2|Du|2 +W (u)

)dx

let us compute separately the variations given by 12 |Du|

2and W (u). As

“Ω∗

1

2|Du|2 dx =

∑a∈Γ∩Ω∗

1

2|Du(a)|2 da

is a quadratic form, for v ∈ V 0 (Ω) we have that(d

du

)∗(“Ω∗

1

2|Du|2 dx

)[v] =

“Ω∗DuDvdx =

“Ω∗

(−∆u · v) dx.

The variation given by W (u) for v ∈ V 0 (Ω) is(d

du

)∗(“Ω∗

(W (u)dx)

)[v] =

(d

du

)∗ ∑a∈Γ∩Ω∗

W (u(a))v(a)da =

“Ω∗W ′(u(x))v(x)dx.

Therefore the total variation of E is

dE(u) [v] =

“Ω∗

(−∆u+W ′(u))) v dx,

which proves our thesis.

The existence of a ultrafunction solution of problem SP follows from the following Theorem:

Lemma 5.5. The functional (5.13) has a minimizer.

Proof. The functional E(u) is coercive in the sense that for any c ∈ R∗

Ec :=u ∈ Vg(Ω) | E(u) ≤ c

is hypercompact (in the sense of NSA) since Vg(Ω) is a hypefinite dimensional affine manifold. Then, sinceE is hypercontinuous (in the sense of NSA), the result follows.

Regarding Ξ being a minimal surface, we can prove the following:


Proposition 5.6. Let u be the ultrafunction minimizer of Problem 5.13 as given by Lemma 5.5. Then thesets

Ω1 = x ∈ Ω | ∀y ∈ mon(x) ∩ Γ, u(y) > 0 ,Ω2 = x ∈ Ω | ∀y ∈ mon(x) ∩ Γ, u(y) < 0

are open, henceΞ :=

x ∈ Ω | ∃y1, y2 ∈ mon(x) ∩ Γ, u(y1) < 0, u(y2) > 0

is closed.

Proof. This follows from overspill8: let us prove it for Ω1. Let x ∈ Ω1. By definition of Ω1, for every ε ∼ 0we have that u(y) < 0 for every y ∈ Bε(x)∩Γ. Hence by overspill there exists a real number r > 0 such thatu(y) < 0 for every y ∈ B∗r (x) ∩ Γ. As Bσr (x) ⊂ B∗r (x) ∩ Γ, we deduce that the open ball Br(x) ⊆ Ω1.

Notice that Property (1) in Proposition 5.6 is a first step towards Property 5.8 in the classical formulationof Problem SP. It is our conjecture, in fact, that Ξ is a minimal surface, at least under some rather generalhypothesis. We have not been able to prove this yet, however.

Let us conclude with a remark. When studying problems like Problem 5.13 with ultrafunctions, onewould like to be able to generalize certain properties of elliptic equations based on the maximum principle:for example, one would expect to have the following properties:

(1) Let Ω be a bounded connected open set with smooth boundary and let g be a bounded function.Then if

u = w + ψ

is the canonical splitting of u as given in Definition 3.9, we have that w ∈ L∞ e ψ(x) ∼ 0 for everyx ∈ Ω;

(2) let Ω1,Ω2 be the sets defined in Proposition 5.6. Then in Ω1 ∪ Ω2 we have

−∆w +W ′(w) = 0,

∆ψ(x) ∼ 0;

(3) if a = inf(g) e b = sup(g) and W ′(t) ≥ 0 for all t ∈ R \ (a, b), we have that

a ≤ u(x) ≤ b.However, in the spaces of ultrafunctions constructed in this paper the maximum principle does not hold

directly: this is due to the fact that the kernel of the derivative is, in principle, larger than the space ofconstants. This problem could be avoided by modifying the space of ultrafunctions: as this leads to sometechnical difficulties, we prefer to postpone this study to a future paper.

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Electronic Journal of Differential Equations, Monograph 07, 2006, (213 pages). ISSN: 1072-6691.

(M. Squassina) Dipartimento di Matematica e Fisica

Universita Cattolica del Sacro Cuore

Via dei Musei 41, I-25121 Brescia, ItalyE-mail address: [email protected]

(V. Benci) Dipartimento di MatematicaUniversita degli Studi di Pisa

Via F. Buonarroti 1/c, 56127 Pisa, Italy

and Centro Linceo interdisciplinare Beniamino SegrePalazzo Corsini - Via della Lungara 10, 00165 Roma, Italy

E-mail address: [email protected]

(L. Luperi Baglini) Faculty of Mathematics

University of Vienna

Oskar-Morgenstern-Platz 1, 1090 Wien, AustriaE-mail address: [email protected]

generalized solutions of variational problems and...

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